Simplify and expand the following expression: $ \dfrac{2}{2p - 20}- \dfrac{4}{4p - 32}- \dfrac{5p}{p^2 - 18p + 80} $
Answer: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $2$ out of denominator in the first term: $ \dfrac{2}{2p - 20} = \dfrac{2}{2(p - 10)}$ We can factor a $4$ out of denominator in the second term: $ \dfrac{4}{4p - 32} = \dfrac{4}{4(p - 8)}$ We can factor the quadratic in the third term: $ \dfrac{5p}{p^2 - 18p + 80} = \dfrac{5p}{(p - 10)(p - 8)}$ Now we have: $ \dfrac{2}{2(p - 10)}- \dfrac{4}{4(p - 8)}- \dfrac{5p}{(p - 10)(p - 8)} $ The least common multiple of the denominators is: $ 8(p - 10)(p - 8)$ In order to get the first term over $8(p - 10)(p - 8)$ , multiply by $\dfrac{4(p - 8)}{4(p - 8)}$ $ \dfrac{2}{2(p - 10)} \times \dfrac{4(p - 8)}{4(p - 8)} = \dfrac{8(p - 8)}{8(p - 10)(p - 8)} $ In order to get the second term over $8(p - 10)(p - 8)$ , multiply by $\dfrac{2(p - 10)}{2(p - 10)}$ $ \dfrac{4}{4(p - 8)} \times \dfrac{2(p - 10)}{2(p - 10)} = \dfrac{8(p - 10)}{8(p - 10)(p - 8)} $ In order to get the third term over $8(p - 10)(p - 8)$ , multiply by $\dfrac{8}{8}$ $ \dfrac{5p}{(p - 10)(p - 8)} \times \dfrac{8}{8} = \dfrac{40p}{8(p - 10)(p - 8)} $ Now we have: $ \dfrac{8(p - 8)}{8(p - 10)(p - 8)} - \dfrac{8(p - 10)}{8(p - 10)(p - 8)} - \dfrac{40p}{8(p - 10)(p - 8)} $ $ = \dfrac{ 8(p - 8) - 8(p - 10) - 40p} {8(p - 10)(p - 8)} $ Expand: $ = \dfrac{8p - 64 - 8p + 80 - 40p}{8p^2 - 144p + 640} $ $ = \dfrac{16 - 40p}{8p^2 - 144p + 640}$ Simplify: $ = \dfrac{2 - 5p}{p^2 - 18p + 80}$